Joint measurability in nonequilibrium quantum thermodynamics

In this Letter we investigate the concept of quantum work and its measurability from the viewpoint of quantum measurement theory. Very often, quantum work and fluctuation theorems are discussed in the framework of projective two-point measurement (TPM) schemes. According to a well-known no-go theorem, there is no work observable which satisfies both (i) an average work condition and (ii) the TPM statistics for diagonal input states. Such projective measurements represent a restrictive class among all possible measurements. It is desirable, both from a theoretical and experimental point of view, to extend the scheme to the general case including suitably designed unsharp measurements. This shifts the focus to the question of what information about work and its fluctuations one is able to extract from such generalized measurements. We show that the no-go theorem no longer holds if the observables in a TPM scheme are jointly measurable for any intermediate unitary evolution. We explicitly construct a model with unsharp energy measurements and derive bounds for the visibility that ensure joint measurability. In such an unsharp scenario a single work measurement apparatus can be constructed that allows us to determine the correct average work and to obtain free energy differences with the help of a Jarzynski equality.

Figure 1

In a general TPM scheme a system is prepared in a state $\rho$ and measured by an initial measurement $\mathsf{A}$. This first measurement is implemented by an instrument ${\mathcal{I}}^{\mathsf{A}}$ that outputs a postmeasurement state ${\rho }_a^{\prime }$ depending on the outcome $a$. After an arbitrary unitary evolution $U$ a final measurement $\mathsf{B}$ is implemented and outcome $b$ is obtained. For each outcome $a$ and $b$, an energy estimate $f\left(a\right)$ and $g\left(b\right)$ is assigned and a work value $w(a,b)=g\left(b\right)-f\left(a\right)$ can be determined. In the case of projective measurements, the estimates $f\left(a\right)$ and $g\left(b\right)$ can uniquely be identified with the energy eigenvalues $E_a$ and $E_b^{\prime }$. For general POVMs such a correspondence is missing. We investigate whether such a scheme can be replaced by a single observable ${\mathcal{W}}^U$ that (i) reproduces the correct average work for any input state as well as (ii) the full statistics for input states diagonal in the initial energy eigenbasis of the system (see main text for precise formulations). For the projective case it is known that such an observable cannot exist. In contrast, if $\mathsf{A}$ and $\mathsf{B}$ are jointly measurable POVMs for any $U$, the answer can be positive.